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re: Rotations` (again)

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  • Subject: re: Rotations` (again)
  • From: wmm at (Martin McClain)
  • Date: Tue, 28 Dec 93 10:17:10 EST

Dear Euler fanciers: 
         Here is a notebook that pins down the WRI Eulere angle convention,
as embodied in Geometry`Rotations`.  This package DEFINITLY needs revision,
but I still cannot recommend a consistent fix.

	        The conventions of the standard package Geometry`Rotations`

In[2]:=  <<Geometry`Rotations`

In[3]:=  ?Geometry`Rotations`*
Rotate2D         Rotate3D         RotationMatrix2D RotationMatrix3D

	Proof that the WRI angle convention is LEFT handed

In[4]:=  RotationMatrix2D[a]
Out[4]=  {{Cos[a], Sin[a]}, {-Sin[a], Cos[a]}}

Look at what RotationMatrix2D does to unit vector ex:

In[5]:= RotationMatrix2D[Pi/2].{1,0} -^-  Out[5]={0, -1}

Vector ex becomes -ey, a _clockwise_ rotation by Pi/2.  
This is Wigner's convention (Wigner, Group Theory, Eq. 14.2, page 143).  
If you hold your _LEFT_ hand with the thumb pointing in the +z direction, 
the fingers curl in the direction of a positive rotation in the x,y plane.

See A. R. Edmonds, Angular Momentum in Quantum Mechanics, pages 6-8, for a 
diagram showing the Euler angles visualized as rotations
about MOVING axes, and a proof that these rotations are exactly equivalent 
to three successive rotations about two perpendicular FIXED axes.

                The convention behind RotationMatrix3D
Define an Euler matrix which implements, in left handed convention, 
the following rotations: 
(1) by a about Z, 
(2) by b about X, 
(3) by g about Z.  

We call it LeftZgXbZa, and it dots into a vector to its 
right because a, the first angle,  is in the rightmost matrix.

In[6]:=  LeftZgXbZa = 

{{ Cos[g],Sin[g],0},
 {    0  ,  0   ,1}}.
{{1,   0,     0   },
 {0, Cos[b],Sin[b]},
{{ Cos[a],Sin[a],0},
 {    0  ,  0   ,1}};

We now show that this is RotationMatrix3D in the standard Mma package 

In[7]:=  LeftZgXbZa ==RotationMatrix3D[a,b,g]
Out[7]=  True

	               	       The Edmonds Euler matrix
Edmonds own Euler matrix, allegedly consistent with the Condon and 
Shortley angular momentum conventions, (but which I have not so far been
to prove consistent, or to pin down an error) is

In[8]:=  EdmondsPage8 = RightZaYbZg = 

 {Sin[a], Cos[a],0},
 {  0   ,   0   ,1}}.
{{ Cos[b],0,Sin[b]},
 {  0    ,1,  0   },
  {Sin[g], Cos[g],0},
 {    0  ,   0   ,1}};
 How does this relate to Mma's Euler matrix?

In[9]:=  EdmondsPage8==RotationMatrix3D[-g-Pi/2,b,-a+Pi/2]
Out[9]=  True

Hope this helps-  Martin McClain, Wayne State, Detroit

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